Compact operators with BMO symbols on multiply-connected domains

Roberto Raimondo

Acta Sci. Math. (Szeged) 84:3-4(2018), 643-658
33/2017

Abstract. In this paper we study the problem of the boundedness and compactness of the Toeplitz operator $T_{\varphi }$ on $L_{a}^{2}(\Omega )$, where $\Omega $ is a multiply-connected domain and $\varphi $ is not bounded. We find a necessary and sufficient condition when the symbol is $\mathcal{BMO}.$ For this class we also show that the vanishing at the boundary of the Berezin transform is a necessary and sufficient condition for compactness. The same characterization is shown to hold when we analyze operators which are finite sums of finite products of Toeplitz operators with unbounded symbols.

DOI: 10.14232/actasm-017-283-0

AMS Subject Classification
(1991): 47B35; 47B38

Keyword(s):
Bergman space,
Toeplitz operator,
Berezin transform

Received May 29, 2017 and in final form September 2, 2018. (Registered under 33/2017.)